0
votes
2answers
29 views

Proving the arithmetic mean equals the geometric mean when $a=b$.

Arithmetic mean $a,b \in \mathbb R$ is $A(a,b)=\frac{a+b}{2}$ Geomtric mean $a,b \in\left[0,\infty\right]$ is $G(a,b)=\sqrt{ab}$ I'm trying to prove that $G(a,b)=A(a,b)$ if and only if $a=b$. ...
1
vote
1answer
28 views

If $(y_n(x))_{n \in \mathbb{N}}$ is uniformly convergent, so is $(f(x,y_n(x)))_{n \in \mathbb{N}} \ ?$

Let $f$ be a continuous function defined on $[a,b] \times [c,d]. $ Consider $(y_n(x))_{n \in \mathbb{N}}$ such that it is uniformly convergent on $[e,f] \subseteq [a,b].$ Could anyone advise me on ...
1
vote
2answers
42 views

If a continuous function is positive at a point, it is also positive in some neighborhood of the point [closed]

Suppose that $f:\mathbb{R}^k\to\mathbb{R}^1$ is a continuous function and that $f(x^*)>0$. Show that there is a ball $B=B_\delta(x^*)$ such that $f(x)>0$ for all $x\in B$.
3
votes
3answers
130 views

Convergent or divergent $\sum_{n=1}^{\infty} \frac{e^nn!}{n^n}$?

Any suggestion/hint, not the whole solution, how to determine convergence/divergence of $$ \sum_{n=1}^{\infty}\dfrac{e^n \cdot n!}{n^n} $$ I'm currently stuck.
0
votes
1answer
21 views

Giving restriction on the value of $N$ in $\epsilon-N$ proof.

I'm still working on $\epsilon-N$ proof. If you don't mind is it possible for us to give restriction on the value of $N$ as illustrated by this example: Say after some manipulation of the limit ...
1
vote
1answer
31 views

Choosing the right N in $\epsilon-N$ proof

I'm just a little bit confused in choosing the right $N$ when working on the rough sketch of the proof. Suppose after some algebra we have reached the point where we get this expression, say: ...
6
votes
1answer
74 views

Alternative proof for the fact that a continuous function on a closed interval attains its boundaries.

Let $f:[a,b]\to \mathbb{R}$ be a continuous function. We are interested in showing that $\exists \beta \in [a,b]$, such that $f(\beta) = M$, where M is its upper boundary. I have managed to proof ...
1
vote
3answers
40 views

Considering $\epsilon$ intuitively in limit proof

I'm having rather difficult time in trying to use $\epsilon$ argument appropriately. For example here is my simple $\epsilon$ proof in one question. The question is as follow: Prove if $s_n \geq 0$ ...
2
votes
1answer
54 views

Proof: $(\sup(A) - \epsilon)^n<y<(\sup(A)+\epsilon)^n$

Prop.: let be $y \in \Bbb{R}_{>0}$, $n \in \Bbb{N}_{>0}$, and $A \subseteq \Bbb{R}$, then: $$A=\{x| x \in \Bbb{R}_{>0}\wedge x^n \leq y \} \Rightarrow (\sup(A) - \epsilon)^n< ...
3
votes
1answer
70 views

Every uncountable subset of $\mathbb{R}$ has a limit point

I am looking at this problem and I decided to attack it by proving the contrapositive. If $E \subset \mathbb{R}$ has finitely limit points, then $E$ is countable. My proof: Let ...
0
votes
3answers
92 views

Proving Riemann Sums via Analysis

Exercise $\bf 5.1.7$: Suppose $f:[a,b]\to\Bbb R$ is Riemann integrable. Let $\epsilon\gt0$ be given. Then show that there exists a partition $P=\{x_0,x_1,\ldots,x_n\}$ such that if we pick any set ...
0
votes
2answers
86 views

Prove that a function is bijective

So, the problem sounds like this. You have two bijective functions $f:\mathbb{N} \to A$, $g:\mathbb{N} \to B$. We define the function $ h:\mathbb{N} \to A \cup B $, defined as: $$ h(n) = ...
4
votes
6answers
551 views

Intro to Real Analysis

I am having trouble proving the following: if $a < b$, then $a < {a+b\over2} < b$. I started with the Trichotomy Property and getting to where $a^2>0$, but then I do not know where ...
2
votes
2answers
51 views

Question about the density of Q in R

So I was looking over a density that shows that the rational numbers are dense in the real numbers. If $0< a <b$, with with $a,b$ real numbers, then I understood why we can chose n such that: ...
0
votes
0answers
26 views

U-substitution proof by partitions

I am trying to prove integration by substitution, i.e. for $f:[c,d] \rightarrow \mathbb{R}$ continuous and $\phi: [c,d] \rightarrow [a,b]$ continuous on $[c,d]$ and $C^1$ on $(c,d)$. Then define ...
1
vote
2answers
41 views

Show a double-sided infinite integral of $\sin(x+b)$ exists iff $b=n\pi$

More formally: Show that $$\lim_{a\rightarrow \infty} \int_{-a}^a \sin(x+b)$$ exists if and only if $b=n\pi$ for some $n \in \mathbb{Z}$. I get the intuition fine. The function is just a horizontal ...
4
votes
0answers
74 views

Proving u-substitution the hard way — use only definition of integration with partitions

I am trying to prove integration by substitution, i.e. for $f:[c,d] \rightarrow \mathbb{R}$ continuous and $\phi: [c,d] \rightarrow [a,b]$ continuous on $[c,d]$ and $C^1$ on $(c,d)$. Then define ...
0
votes
2answers
55 views

Ideas on how to proceed with a proof?

Sorry if this is a nonspecific question - I can provide more details but at this point I need general ideas on a proof strategy. So I recently reduced a rather difficult optimization problem to ...
3
votes
2answers
76 views

Passing a derivative through a limit.

After searching around on the net and on SE I have not found a satisfactory answer. Let $f_n: D \to \mathbb R$ be a sequence of functions. What assumptions, aside from $f$ being differentiable, do we ...
23
votes
2answers
2k views

Single Variable Integral

Compute the following integral \begin{equation} \int_0^{\Large\frac{\pi}{4}}\left[\frac{(1-x^2)\ln(1+x^2)+(1+x^2)-(1-x^2)\ln(1-x^2)}{(1-x^4)(1+x^2)}\right] x\, ...
0
votes
3answers
47 views

Strategy to solve inequalities

I want to prove $|x + y| = |x| + |y|$ iff $xy \ge 0$. I don't understand a place to start. I was thinking of solving using contradiction but, as I am new to real analysis, I don't understand it. ...
0
votes
0answers
32 views

Show if this is integrable (defined 1 on rationals, 0 else)

Define $f: [0,1] \rightarrow \mathbb{R}$ as $f(x) = \begin{cases} 0 & x \in \mathbb{Q} \\ 1 & x \notin \mathbb{Q} \end{cases}$ Find $\underline{\int_0^1f}$ and $\overline{\int_0^1f}$. Is ...
1
vote
1answer
14 views

Limit of a function proof verification

My proof: By Bernoulli Equation $(a^n+b^n)^{1/n}=b(1+(na)/b)^{1/n}$ By definition of a limit, fix $\epsilon > 0$ and $N>(b\epsilon^n)/a$ Then, $|a_n - b | = ...
1
vote
1answer
24 views

Inequalities involving x and y.

I am asked to prove: $(x-y)^3 \ge x^3-3x^2y$ where $x,y$ are real and $0 < y < x$ I am told Bernoulli's inequality may help. I have however reduced this to $3xy^2 - y^3 \ge 0$. I have ...
2
votes
3answers
72 views

Proving analytical statement, Intermediate Value Theorem

Let's define $f$ as a continuous function with $f:[0;2] \to \mathbb{R}$ and $f(0) = f(2)$. Now, I want to show that: $$\exists x_0 \in [0;1]:f(x_0) = f(x_0 + 1)$$ I tried to plot a few functions in ...
6
votes
2answers
221 views

Proof of continuity - (ε-δ) definition - Can anyone check this?

I've been trying to get my head around this problem for quite some time by now. I want to prove that $$f(x) := \left|\frac{x-1}{x^2+1}\right|$$ is continuous for $$x_0 = -1$$ Now, in order to prove ...
2
votes
2answers
30 views

Two sequences are equivalent. Prove that one is Cauchy iff the other is Cauchy.

This question has already been asked and answered here Let $ϵ>0$ be given. With loss of generality, we may assume $ϵ$ is rational. Suppose $a_n$ is a Cauchy sequence and $b_n, a_n$ are ...
2
votes
0answers
49 views

Continuity of the right-hand derivative of a Convex function (help with the proof)

Hi everyone I have some trouble with one point in the following proof. Let $f$ be a convex function (strict convex function) on a real interval. If $f'_-(a)=f'_+(a)$ where $f'_-$ and $f'_+$ are ...
2
votes
1answer
221 views

A sequence converges $\iff$ it's Cauchy. Proof of ($\Leftarrow$) (Abbott p 59 t2.6.4)

Lemma 2.6.3 $\implies (x_{n})$ is bounded. So use the Bolzano-Weierstrass Theorem to produce a convergent subsequence $(x_{n_{k}})$ . Set $x= \lim x_{n_{k}}.$ So $(x_{{n_{k}}}) \to x. \quad ...
1
vote
2answers
106 views

Cauchy Sequences are Bounded. Questions on proof (Abbott p 59 lemma 2.6.3)

By the agnecy of p 44 Definition 2.3.1, we find M > 0 such that $|a_n| \le M$ for all $n \in N$. Not duplicate. Proof. By definition, given any $e >0$ there is an integer $N$ such that $|a_{n}- ...
2
votes
2answers
53 views

Contraction Mapping Theorem. Any $\{ x,f(x),f(f(x)),\ \ldots) \} $ converges to the unique fixed point of f. (Abbott p 114 q4.3.9 d)

Let $f$ be a function defined on all of $R$. Assume there is a constant $c$ such that $0< c <1$ and $ |f(x)\ -f(y)\leq c|x-y|$ for all $x,\ y\in R$. Parts a,b. (c) Prove that $y$ is a fixed ...
1
vote
1answer
96 views

Contraction Mapping Theorem. Prove $\{ y_{1},f(y_{1}),f(f(y_{1})),\ \ldots) \} $ is Cauchy. (Abbott p 114 q4.3.9b)

Let $f$ be a function defined on all of $R$. Assume there is a constant $c$ such that $0< c <1$ and $ |f(x)\ -f(y)\leq c|x-y|$ for all $x,\ y\in R$. (a) Show that $f$ is continuous on $R$ for ...
2
votes
0answers
38 views

Continuous Functions map compact sets to compact sets. Modus operandi of proof (Abbott p 115 t4.4.2)

Theorem 4.4.2 (Preservation of Compact Sets). Let : $A\rightarrow R$ be continuous on A. If $K\subseteq A$ is compact, then.$f(K)$ is compact as well. p 84 Definition 3.3.1: A set $K \subseteq R$ is ...
1
vote
2answers
74 views

if $g$ is continuous at $c$ and $g(c)\neq 0$, there exists an open interval containing $c$ on which $f(x)/g(x)$ is defined (Abbott p 113 q4.3.5)

Theorem 4.3.4.(iv) says that $f(x)/g(x)$ is continuous at $c$ if both $f$ and $g$ are, provided that the quotient is defined. Show that if $g$ is continuous at $c$ and $g(c)\neq 0$, then there exists ...
0
votes
3answers
60 views

Sequence that contains subsequences converging to every point in the infinite set $\{1/n \} \, \forall \, n \in N$ (Abbott p 58 q2.5.3c)

has this property. Notice that there is also a subsequence converging to 0. We shall see that this is unavoidable. I acquiesce to this example, but I wasn't conscious of it until I read ...
4
votes
1answer
305 views

If every convergent subsequence converges to $a$, then so does the original bounded sequence (Abbott p 58 q2.5.4 and q2.5.3b)

Assume $(a_{n})$ is a bounded sequence such that every convergent subsequence of $(a_{n})$ converges to the same limit $a\in \mathbb{R}$. Show $(a_{n})$ must converge to $a$. Prove by contradiction. ...
3
votes
2answers
53 views

A subsequence of a convergent sequence converges to the same limit. Questions on proof. (Abbott p 57 2.5.1)

Solutions to Homework 3 doesn`t duplicate. We have to prove that if $(a_{n})$ is a sequence in $\mathbb{R}$ with $\displaystyle \lim_{n\rightarrow\infty} a_n =a$, and if $(a_{n_{k}})_{k\in ...
2
votes
1answer
48 views

Prove shuffled sequence $\{x_i, y_i\}$ converges $\iff \lim x_n = \lim y_n$ (Abbott p 49 q2.3.5)

Let $(x_{n})$ and $(y_{n})$ be given. Define $(z_{n})$ to be the shuffled sequence $(x_{1}.y_{1},\ x_{2},\ y_{2},\ x_{3},\ldots,x_{n}, y_{n},\ldots)$ . Prove that $(z_{n})$ is conv ergent $\iff ...
0
votes
2answers
40 views

Integrable functions and absolute values

I have qutoted that the absolute value of an integral is less than or equal to the integral of an absolute value of a function. I have also said $|-g(x)| \le g(x) \le |g(x)|$ implies the integral ...
1
vote
1answer
34 views

Show that we can reorder mixed partials, if every partial is continuous

Suppose $f$ has all partial derivatives up to and including $k$ and all of these partials are continuous. Prove that if $\sigma$ is a permutation on $n$ letters (any reordering), then ...
2
votes
2answers
45 views

Help with real analysis proof involving supremum

Let $S\subseteq\Re$ be nonempty. Prove that if a number $u$ in $\Re$ has the properties: (i) for every $n\in N$ the number $u-1/n$ is not an upper bound of $S$, and (ii) for every number $n\in N$, the ...
0
votes
2answers
33 views

How to quantify this statement

How do I quantify this statement? Let $x \in \mathbb{R}$. $1 \le x \le 2$ if and only if $1 \le x \le 1+ 1/n$ for some $n \in \mathbb{N}$. When I am trying to prove this, I am led (in the course of ...
1
vote
1answer
69 views

$\epsilon - N$ proof confirmation.

These proofs seem to be my absolute worst problem. I just don't seem to get them, that being said, if this is right, I may have started to get the hang of it. My limit and required assumptions: ...
0
votes
0answers
10 views

prove where $f(x,y) = \sqrt{|x| + |y|}$ is differentiable and is not differentiable [duplicate]

Classify where $f(x,y) = \sqrt{|x| + |y|}$ is differentiable -- where it's not, prove it, and where it is, prove it. For $x\neq 0$ and $y\neq 0$, we can just treat it as $f(a,b) = \sqrt{a+b}$ (where ...
0
votes
1answer
42 views

Suppose that the sequence of prices{$p_k$} converges to a limiting price $\bar p$. What must $\bar p$ be?

We let $Q_k$ denote the supply of commodity, $D_k$ the demand for the commodity, and $p_k$ the price at $k$-th time. The demand depends on the current price, $D_k = a + b p_k$ and the supply depends ...
0
votes
1answer
45 views

General conceptual confusion relating to vacuous proofs and quantifier help

I need to prove the statement: Let $x \in \mathbb{R}$. Prove that $1 \le x \le 2$ if and only if $1 \le x \le 1+ 1/n$ for some $n \in \mathbb{N}$. So I start with the forward implication: If $1 ≤ ...
1
vote
1answer
42 views

We are interested in price of a commodity, traded at regular intervals. Why it is reasonable to take $a$, $c$, and $d > 0$ and $b < 0$?

We are interested in the price of a commodity which is traded at regular intervals. We let $Q_k$ denote the supply of commodity, $D_k$ the demand for the commodity, and $p_k$ the price at $k$-th time. ...
4
votes
1answer
117 views

A tough one: show that this is not differentiable at any point in R

Here's the question: Define $\phi: \ \mathbb{R} \rightarrow \mathbb{R}$ by $$ \phi(x) = \begin{cases}x & 0\leq x\leq\frac{1}{2}\\ 1-x & \frac{1}{2}\leq x\leq 1\end{cases}. $$ And then ...
0
votes
1answer
74 views

show that $f(x,y) =2x^2 + 3y$ is differentiable at $(0,0)$ by finding a linear function T

Here's the question: Prove that $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ defined $f(x,y) = 2x^2 + 3y$ is differentiable at $\begin{bmatrix} 0\\0 \end{bmatrix}$ by producing a linear function T and ...
1
vote
4answers
76 views

Prove that the set of triples $\{(a,b,c)|a,b,c \in \mathbb{N}\}$ is countable

I have the following question in my textbook: Prove that the set of triples $\{(a,b,c)|a,b,c \in \mathbb{N}\}$ is countable Now I know that $\mathbb{N}$ is countable already, and I have completed a ...